A refinement of theorems on vertex-disjoint chorded cycles

In 1963, Corrádi and Hajnal settled a conjecture of Erdős by proving that, for all \(k \geq 1\), any graph \(G\) with \(|G| \geq 3k\) and minimum degree at least \(2k\) contains \(k\) vertex-disjoint cycles. In 2008, Finkel proved that for all \(k \geq 1\), any graph \(G\) with \(|G| \geq 4k\) and minimum degree at least \(3k\) contains \(k\) vertex-disjoint chorded cycles. Finkel's result was strengthened by Chiba, Fujita, Gao, and Li in 2010, who showed, among other results, that for all \(k \geq 1\), any graph \(G\) with \(|G| \geq 4k\) and minimum Ore-degree at least \(6k-1\) contains \(k\) vertex-disjoint cycles. We refine this result, characterizing the graphs \(G\) with \(|G| \geq 4k\) and minimum Ore-degree at least \(6k-2\) that do not have \(k\) disjoint chorded cycles.